TIME-SERIES AND DEPENDENT-VARIABLES

We present a new method for analyzing time series which is designed to extract inherent deterministic dependencies in the series. The method is particularly suited to series with broad-band spectra such as chaotic series with or without noise. We derive quantities, delta-j(epsilon), based on conditional probabilities, whose magnitude, roughly speaking, is an indicator of the extent to which the k th element in the series is a deterministic function of the (k - j)th element to within a measurement uncertainty, epsilon. We apply our method to a number of deterministic time series generated by chaotic processes such as the tent, logistic and Henon maps, as well as to sequences of quasi-random numbers. In all cases the delta-j correctly indicate the expected dependencies. We also show that the delta-j are robust to the addition of substantial noise in a deterministic process. In addition, we derive a predictability index which is a measure of the extent to which a time series is predictable given some tolerance, epsilon. Finally, we discuss the behavior of the delta-j as epsilon approaches zero.